An electrostatic actuation oscillator of the electromechanical microsystem or nanosystem type, respectively MEMS (“microelectromechanical system”) or NEMS (“nanoelectromechanical system”), is a device including a mobile component which can be set to oscillate.
Such oscillators consume little energy and have a reduced size. They therefore have particularly advantageous applications in roaming objects, such as smartphones or digital tablets. These oscillators are used in particular to manufacture inertial sensors, such as accelerometers or gyroscopes, intended to be fitted into such objects.
A roaming object can also be fitted with a satellite location system (GPS for “Global Positioning System”) which consumes a lot of energy. Inertial sensors enable the movements of the object to be computed in real time, and therefore enable use of the satellite location system to be reduced. This latter system nonetheless remains useful for determining a reference position of the object, for example at regular time intervals.
The electrostatic actuation oscillator contains actuation means which transform an excitation signal applied at the input of the oscillator into an electrostatic actuation force, also called an “excitation force”, acting on the mobile component. The movements of the mobile component are measured by detection means which generate a response signal at the output of the oscillator. The excitation signal and the response signal can take the form, for example, of voltages.
The oscillator is characterised by two essential parameters, namely a resonant frequency and a quality factor. The oscillations of the mobile component are indeed governed by motion equations which are modified when an external force, for example due to acceleration or a gyroscopic effect, is exerted on the mobile component. The resonant frequency is then also modified. If the oscillator's initial resonant frequency is known, it is therefore possible to quantify the applied external force.
It is also preferable to excite the oscillator to a frequency close to the resonant frequency so as to maximise the amplitude of the oscillations. By this means the force being exerted on the mobile component can be detected more easily, and the sensor's sensitivity is improved.
The oscillator's quality factor, for its part, influences the accuracy of the inertial sensor. The higher the quality factor the more accurate the sensor, and the more it is possible, for example, to limit use of the satellite location system.
It is therefore important to measure the oscillator's resonant frequency and quality factor, in particular to validate the development and manufacture of the inertial sensor.
A first method of measuring the resonant frequency and quality factor consists in exciting the oscillator by means of a sinusoidal excitation voltage of frequency F0 and in measuring the amplitude of the oscillations at frequency 2F0. The amplitude of the oscillations is measured at frequency 2F0 since the electrostatic actuation force is proportional to the square of the excitation voltage. By sweeping a frequency range an amplitude spectral density of the oscillations is obtained, from which the resonant frequency can be determined. Indeed, the amplitude spectral density forms a resonance peak with a maximum amplitude attained at the resonant frequency. The quality factor can, for its part, be determined from the breadth of the resonance peak considered at half its height.
One disadvantage of this method of measuring by frequency sweep is that in order to measure the amplitude of the oscillations correctly the oscillator must return to an idle position between two successive measurements. The time required for the oscillator to return to its idle position after excitation is approximately equal to three times a damping constant which is proportional to the quality factor. If one considers, for example, a quality factor of the order of 106, the damping constant is of the order of 10 seconds. In this case over 2 hours are required to acquire 250 measuring points.
Furthermore, the accuracy with which the resonance peak is defined also depends on the number of measuring points. If the number of points is not sufficient, the resonance peak can be broadened artificially and its maximum amplitude can be frequency shifted, which distorts the measurement of the resonant frequency and of the quality factor.
The time required to implement this measuring method is therefore a factor which limits productivity, particularly since the current trend is to increase the quality factor of oscillators. This measuring time does not reasonably enable, in particular from an industrial standpoint, to characterize all the oscillators of a silicon plate, which habitually contains over one hundred of them. Only a few oscillators are then characterized, which makes it difficult to estimate an oscillator manufacturing yield.
A second method of measuring the resonant frequency and the quality factor, called the “broadband pulse” method, consists in exciting the oscillator by means of a voltage pulse and in acquiring the response signal generated by the oscillator. Indeed, a pulse in the time domain corresponds to a constant amplitude in the frequency domain. In practice the pulse tends to have a high spectral bandwidth. Consequently, by calculating the Fourier transform of the acquired signal, it is possible to locate the resonance peak present in the pulse's spectral band. This is a rapid measuring method since the acquisition can last only a few seconds.
Using this method, the acquisition duration directly determines the frequency resolution of the measurement. For example, an acquisition duration of 1 second gives a resolution of 1 Hertz, and an acquisition duration of 100 seconds gives a resolution of 0.01 Hertz.
One disadvantage of this measuring method by broadband pulse is that the electrostatic actuation oscillator has a parasitic capacity between the actuation means and the detection means. There is therefore direct electrical coupling of the excitation voltage towards the oscillator's output. Since the excitation voltage is broadband its amplitude spectral density is then superimposed on the amplitude spectral density of the response signal. If the capacitive coupling is sufficiently large a weak oscillation of the oscillator is then immersed in the spectrum of the excitation voltage, which generally makes this measuring method unusable.
To overcome this problem, instead of a broadband pulse it is possible to use a narrowband pulse to excite the oscillator. This narrowband pulse is shaped, for example, like a cardinal sine function of spectral width ΔF modulated by a sine function of frequency F0. Excitation voltage u(t) can then be defined by the following equation:u(t)=sinc(ΔF·t)·sin(2·π·F0·t)  (1)
The excitation force is proportional to the square of excitation voltage u(t) and can be expressed by the following equation:F(t)=k·[sinc(ΔF·t)·sin(2·π·F0·t)]2  (2)where k is a proportionality factor. Equation (2) can be developed to obtain the following equation:F(t)=k[½·sinc2(ΔF·t)−½·cos(4·π·F0·t)·sinc2(ΔF·t)]  (2′)The excitation force is therefore the sum of a first term proportional to the square of the cardinal sine and of a second term proportional to the square of the cardinal sine modulated at frequency 2F0.
FIG. 1 shows the amplitude spectral density (DSA) 110 of the excitation voltage u(t). This spectral density 110 is a rectangle function of breadth ΔF centred on frequency F0. FIG. 1 also shows the amplitude spectral density of excitation force F(t) which includes a first component 121 and a second component 122 corresponding respectively to the first term and the second term of equation (2′). Each of these spectral components 121, 122 has a triangular shape of width 2ΔF, where first component 121 is centred on 0 Hz and second component 122 is centred on 2F0.
The resonant frequency of the oscillator is sought in a zone which corresponds to the frequency range over which second component 122 extends. By appropriately defining frequency F0 of the sine and spectral width ΔF of the cardinal sine, the amplitude spectral density 110 of the exciting pulse transmitted by capacitive coupling is not superimposed with the amplitude spectral density of the electrostatic actuation force, as illustrated in FIG. 1. Thus, the amplitude spectral density resulting from the capacitive coupling of the exciting pulse is not present in the zone where the resonant frequency is sought.
One disadvantage of this measuring method by narrowband pulse is that the amplitude spectral density of the excitation force is not constant. It is, indeed, at a maximum at frequency 2F0, and decreases linearly on both sides, until it is cancelled at frequency 2F0−2ΔF and at frequency 2F0+2ΔF. The oscillator is therefore excited with a force which depends on the difference between the resonant frequency and frequency 2F0. This phenomenon is shown in FIGS. 2A and 2B.
FIG. 2A shows three examples of amplitude spectral densities 211, 212, 213 of the excitation force when frequency F0 is equal respectively to 16.25 kHz, 16.5 kHz and 17.5 kHz. The resonant frequency of the oscillator is sought in the frequency window formed by each of these spectral densities To each of spectral densities 211, 212, 213 of the excitation force corresponds an amplitude spectral density 221, 222, 223 of the response signal acquired at the output of the oscillator.
FIG. 2B is an enlargement of FIG. 2A around the resonance peak. It can be observed that the closer this peak is to the edge of the frequency window the lower its amplitude. The shape of the amplitude spectral density of the electrostatic force can therefore lead to conclude that the oscillator is not functional, when it is only that the excitation force is too weak to cause the oscillator's mobile component to start oscillating.